Anti-aliasing of spatial frequencies by geophone and source placement

ABSTRACT

In the generation, recording, and processing of seismic wave data, coherent noise waves of higher spatial frequencies (shorter wave lengths) than those in the desired and useful spatial frequency pass band tend to &#39;&#39;&#39;&#39;alias&#39;&#39;&#39;&#39; down into the desired pass band and to produce interfering effects. This disclosure tells how improved arrays of sources and receivers may be designed and used that, instead of attempting to suppress all spatial frequencies higher than those in the desired pass band, suppress preferentially only those particular, higher spatial frequencies that would alias back into the desired pass band.

United States Patent [191 Muir et a1.

[ 1 March 6, 1973 I54] ANTI-ALIASING OF SPATIAL FREQUENCIES BY GEOPHONEAND SOURCE PLACEMENT [75] Inventors: Francis Muir, Huntington Beach;

- Jerry L. Morrison, Hacienda Heights, both of Calif. [73] Assignee:Chevron Research Company, San

Francisco, Calif.

[22] Filed: Sept. 3, 1971 [21] Appl. No.: 177,723

[52] US. Cl 340/155 AG, 340/155 CP, 340/155 MC [51] Int. Cl. ..G0lvl/16, GOlv 1/28 [58] Field of Search.340/15.5 CP, 15.5 MC, 15.5 AP

[56] References Cited UNITED STATES PATENTS 3,400,783 9/1968 Lee..340/l5.5

3,096,846 7/1963 Savit et al ..340/l5.5

3,400,783 9/1968 Lee ..181/.5

OTHER PUBLICATIONS Schoenberger, Optimization and Implementation ofMarine Seismic Arrays, 12/70, pg., 1038-1053 Geophysics, Vol. XXXV, No.6.

Primary Examiner-Samuel Feinberg Assistant Examiner-N. MoskowitzAttorneyJ. A. Buchanan, G. F. Magdeburger, R. L. Freeland, Jr. and H. D.Messner [5 7] ABSTRACT In the generation, recording, and processing ofseismic wave data, coherent noise waves of higher spatial frequencies(shorter wave lengths) than those in the desired and useful spatialfrequency pass band tend to alias down into the desired pass band and toproduce interfering effects. This disclosure tells how improved arraysof sources and receivers may be designed and used that, instead ofattempting to suppress all spatial frequencies higher than those in thedesired pass band, suppress preferentially only those particular, higherspatial frequencies that would alias back into the desired pass band.

16 Claims, 13 Drawing Figures TRUE HORIZONTAL WAVELENGTH A9 TRUEWAVELENGTH IN /a, DIRECTION OF TRAVEL c; 5 C

PATENTEUMAR 61975 SHEET 10F 8 ATToRflEYs PATENTEDHAR 61973 3,719,924

SHEET 2 UF 8 I I I I M 9 3/Ag /Ag I/Ag o 1/Ag g g F l G 2 A INVENTORSFRA/vc/s M JERRY L. MORRISON ATTORNEYS rmmeom 61m SHEET 0F 8 SPATIALFREQUENCY-P X 1O' (CYCLES/FOOT) n1 2. mmZOmmmm mm FIG.4B

ARRAY CENTER w H H 5 4 5 a k w 5 3 LP F |G.4A

INVENTORS FRANCIS MU/R JERRY L. MORRISON ATTORNEYS PATENTEUHAR 61975SHEET 5 OF 8 N l m mmdl s m RRRQ v E TmAN v NM. v, u mmw mm ATTORNEYSPATENTEDHAR 6197s SHEET 6 OF 8 LL! J J! SPATIAL FREQUENCY-P9 x1O2(CYCLES/FOOT) m m 1 a 2 z. mwzommum mm l l l l l l l I I n c a o I ao u o a u w COMMON GEOPHONE GATHER FIG.7

INV ENTO RS FRANCIS MU/R JERRY L. MORRISON BY M A 'jlw ATTORNEYSPATENTEDHAR 61973 719,924

SHEET 7 BF 8 FIG.8 INVENTORS FRANCIS MU/R JERRY L. MORE/SON PATENTEDHAR61973 SHEET 8 0F 8 HWIHUWIHEII FIG.9

INVENTORS FRANCIS MU/R JERRY L. MORRISON ATTORNEYS ANTI-ALIASING OFSPATIAL FREQUENCIES BY GEOPHONE AND SOURCE PLACEMENT FIELD OF THEINVENTION The present invention pertains to the generation, recording,and processing of seismic wave data for seismic exploration purposes.More specifically it pertains to the improved design of source arraysand receiver arrays, and to improved ways of compositing the receivedsignals so as to suppress undesired, interfering, alias energy.

STATE OF THE ART In seismic prospecting, elastic waves are created at oradjacent to the surface of the earth by several conventional means.Portions of these elastic waves are transmitted downwardly through theearth and are reflected back to the earths surface from the interfacesof subsurface strata. These reflected waves are detected by placinggeophones on or adjacent to the surface of the earth at points displacedfrom the origin of the seismic energy. The geophones convert thedetected waves to equivalent electrical signals which are then amplifiedand recorded in a form known as a seismogram or seismic record. Theusual seismic record consists of a plurality of amplitude versus timetraces aligned in parallel, each representative of the electrical output"of a geophone (or group of geophones) at a known geographic location.

In addition to the aforementioned downwardly transmitted elastic waves,most seismic sources generate a variety of other types of elastic wavesincluding, but not limited to, direct air waves and surface waves, whichare collectively classified as source-generated coherent noise waves.These coherent noise waves will typically have relatively lowerhorizontal components of velocity (higher moveouts) as measured on aseismogram, than the reflections from the interfaces of deep subsurfacestrata.

In the prior art, one means of attenuating the undesirable coherentnoise waves has been to use arrays or groups, of sources, and arrays ofdetectors, instead of a single source and single detectors. Within thisspecification the term geophone group will be used to identify acollection of detectors which are electrically connected in serieseither actually or effectively, so that a single trace is recordedcorresponding to the group and is equivalent to the summed outputs ofthe individual detectors in the group. The location of the geophonegroup is specified by the location, with respect to some fixedcoordinate system, of a single point associated with the geophone array,and called the geophone group center. In practice the detectors within agroup are usually located symmetrically about the geophone group center.The term source group will be used to identify a collection of sources,either explosive or mechanical, which are activated simultaneously, orsequentially, to produce a single seismogram. If the sources within thesource group are activated sequentially, the resulting traces are latercombined in such a way that the final seismogram is essentiallyequivalent to the seismogram that would have been produced had all ofthe sources within the group been activated simultaneously. The locationof the source group is specified by the location of a single pointassociated with the source group and called the source group center. Inpractice the sources in a source group are usually located symmetricallyabout the source group center.

In addition to the terms defined above we shall have occasion to referto the concept of record time. Within this specification the term recordtime will be used to identify the elapsed time after the instant ofactivation of the source. It is conventional practice to record on asingle seismic record the outputs from 24 or more geophone groups whosegeophone group centers are spaced over distances up to 7,000 feet fromthe source array center. The seismic record also generally includes anadditional trace showing the instant at which the seismic source arraywas activated, generally referred to as a .time break trace, and one ormore constant frequency timing signals which may be correlated with therecorded information on the seismic traces to determine the elapsed timefrom the instant of source activation. The elapsed time to theappearance of some event on the seismogram is the record time of thatevent. The record times of an event will in general be different on twodifferent traces of the same seismogram.

To obtain a single seismogram there are two methods in common use. Inthe first method, usually referred to as end-on recording, the geophonegroup centers are located at equal intervals, called the geophone groupinterval, along an approximately straight line on or near the surface ofthe earth. The source group ceri'ter is located on the same straightline but lies at some offset, normally equal to an integral multiple ofthe geophone group interval, beyond one or the other of the two endgeophone group centers. That is, all of the geophone group centers lieon one side of the source group center. In the other method, usuallyreferred to as split-spread recording, the geophone group centers aresymmetrically located about the source group center. The spacing betweenadjacent geophone group centers is again uniform and is again referredto as the geophone group interval. The source group center is usuallylocated at the mid-point of the gap separating two sets of geophonearrays. If the gap, which is normally an integral multiple of thegeophone group interval, is larger than one geophone group interval, thelayout is referred to as a gapped split spread.

The two methods described above for obtaining seismograms are bothapproximate realizations of an idealized theoretical method in whichthere would be an infinite number of geophone group centers, equallyspaced and lying on a straight line. In this theoretical method thesource group center could be located anywhere on the same straight lineas the geophone group centers, but normally it would be located at thesame location as one of the geophone group centers or midway between twoadjacent geophone group centers.

The above paragraphs have mentioned many of the essential elements ofthe seismic exploration method,

and some of its undesirable concomitants, such as thecoherent noisewaves. There is a considerable literature, in both technical journalsand patents, on these coherent noise waves, and how their effects may besuppressed. Much of that literature is relevant in a general way to themethod of the present invention. Much of it has been concerned with thepurposeful design of geophone arrays to eliminate known, or

completely in the following specification, is briefly, that the spacingof the geophone groups themselves causes particular importance to becomeattached to a particular plurality of narrow frequency bands in thespatial frequency spectrum of the coherent noise waves. The solution tothe problem is not given by the literature solutions of designinggeophone arrays that merely give lowered peak heights in the spatialfrequency pass bands of their geophonegroup-response-versus-spatialfrequency functions. The solution to theproblem is the new one ofinsuring that entire pluralities of the valleys(the zeros) of the geophone group response function coincide as exactlyas possible with the pluralities of the most troublesome bands in thespatial frequency spec trum of the coherent noise waves.

An interesting sidelight on the problem recognized and solved by thepresent invention is that it is mathematically provable that the problemcannot actually be shown to exist by anytype ofexamination of fieldrecords, until it has-been solved. Then, the new records may be comparedwith the previous records to demon strate that the problem existed. Twofigures (FIGS. 8 and 9) attached to the present specification illustratesuch a comparison.

To substantiate the above points as well as to provide desirablereferences as background for the understanding and appreciation of thepresent invention, the most relevant technical articles and patentsknown to the present inventors will be briefly mentioned here:

1949, June: C. G. Dahm. U. S. Pat. No. 2,473,469. The Dahm patent isrelevant to the present invention only in the specific sense that itdescribes a method of combined geophone placement and temporal filteringthat would substantially eliminate particular seismic waves havingindividually specified horizontal wavelengths. The Dahm method would notbe practicable in modern field practice.

1952, January: A. Wolf. U. S. Pat. No. 2,580,636. Wolfs teachings,including his figures, are relevant to the present invention becausethey show field mixing of signals from pluralities of geophone groups,which are in turn composed of pluralities of individual geophones.

1955, January: J. O. Parr. U. S. Pat. No. 2,698,927. Parr teachescombining the outputs of pluralities of geophones, and also pluralitiesof sources, using variable weighting. In some embodiments of the presentinvention the outputs of pluralities of whole groups of geophones, andwhole groups of sources, are combined using binomial weighting. Parrteaches that binomial weighting itself was already prior art as of hisfiling date ofJuly 1953.

I955, July: J. O. Parr, Jr. and W. H. Mayne. A New Method of PatternShotting." Geophysics, XX, No. 3, pp. 539-564. Parr and Mayne teach thatundesirable energy could be eliminated by purposeful regulation ofsource and detector patterns. However, their teaching concerns a singlebroad continuous band of disturbing wavelengths." They talked to termsofmaximum relative effect, which referred to maximum peak values of thegeophone (or source) group response function in the frequency band inquestion. They did not discuss a the valleys between the multiple peakvalues of the geophone (or source) group response functions, nor suggestany possible usefulness of regulating the placement of those valleys.

1956, April: M. K. Smith. Noise Analysis and Multiple SeismometerTheory. Geophysics, XXI, No. 2, pp. 337-360. Smiths article is, at leastmathematically, the most erudite treatment of the subject. Smith streatment is based on the theory of Generalized Harmonic Analysisdeveloped by Norbert Wiener and his associates. According to Smith, histheory is so general that initiallylit requires] no restrictiveassumptions as to the nature of the signal or the noise. At no point- Idoes Smith suggest the existence of the particular problem treated inthe present invention, and he makes no mention of the purposefulplacement of the valleys of the geophone (or source) group responsefunctions or any possible usefulness of those valleys.

1958, January: C. H. Savit, J. T. Brustad, and J. Sider. The MoveoutFilter. Geophysics, XXIII, No. 1, pp. 1-25. Savit et al clarified someof the concepts of the previous authors. They made very clear theconcept moveout filter. Such a filter will pass low moveouts,

that is, low values of spatial frequency, and reject the highermoveouts." They made very clear the generally expectable shape of thegeophone (or source) groupresponse-versus-spatial-frequency function.They taught the concept of a meta-array of identical arrays which isused in the present invention (although not by that name) and theytaught that such an array would have a response curve equal to theproduct of the response curve of the typical [component] array by theresponse curve of the meta-array considered as a simple array." However,like the other mentioned authors, they taught, and they even taughtexplicitly that substantially equal importance is attached to noise ofall spatial frequencies within the range of investigation." It isprecisely because this teaching does not represent seismic reality thatthe problem solved by the present invention arises.

July, 1963: U. S. Pat. No. 3,096,846 was filed in December 1958 with thesame three, above-mentioned authors named as the inventors. Theteachings of the patent specification are essentially the same as thoseof the article. It should be especially noted here that in thespecification occurs the phrase .a least squares best fit to, .zeroresponse over the range of noise moveouts." This should not be confusedwith the actual zero responses at selected frequencies that are taughtin the present specification.

1968, September: E. K. Lee. U. S. Pat. No. 3,400,783. To the knowledgeof the present inventors, this is the patent most closely related to thepresent invention. It deals with purposeful geophone placement to filterout waves of higher spatial frequencies, but it is like theabove-mentioned technical literature references in its treatment of abroad band of higher Spatial frequencies, and it gives no recognitionwhatsoever to the possible use of purposefully placed valleys ofgeophone group response functions. In fact, in the Figures of Lee, thevalleys, which on his logarithmic, decibel plots should extend to minusinfinity, are filled in; and the fact that they should actually be zerosis disregarded in the graphical approximations.

1969, March: J. P. Lindsey. U. S. Pat. No. 3,432,807. Lindseys teachingsshow, as of this relatively late date, the continuing effort to usefiltering and combining of the signals from geophone groups in theattempt to eliminate the entire upper band of spatial frequencies(Lindsey uses the term wave numbers) above a preselected value.

1969, September: R. O. Lindseth, H. J. Hoogstraat, and K. H. Tseng.Application of the Two-Dimensional Fourier Transforms to Enhancement ofSeismic Data. Presented to the Society of Exploration Geophysicists,Calgary, Canada. These authors mention the theoretically necessaryexistence of an aliasing phenomenon when spacing of geophone groups isnot fine enough adequately to sample the shortest horizontal wavelengthstraveling in the earths surface. They give no indication that it is asignificant present field problem and suggest only that closer tracespacing in the field will limit the aliasing problem. Close enoughgeophone group spacing to produce results comparable to those producedby the present invention would not be practicable.

1970, January: R. O. Lindseth. Recent Advances in Digital Processing ofGeophysical Data. CDP Computer Data Processors Ltd. 1370 Calgary House.550 6th Ave. S.W., Calgary 1, Alberta, Canada. Lindseth discusses apossible spatial frequency aliasing problem, and mentions what kinds offalse dips could appear on a record as results of aliasing, but again,as a solution to the possible problem mentions only that geophonegroupings must be spaced closer together." No hint is given of thepossibility of the economically feasible solution provided by thepresent invention.

1970, December: M. Schoenberger. Optimization and Implementation ofMarine Seismic Arrays. Geophysics, XXXV, No. 6, pp. 1038-1053.Schoenberger treats the problem of designing arrays to reject specifiedbands of wavelengths which contain noise. Like the analyses of thepredecessors, Parr and Mayne, Savit et al., and Smith, Schoenbergersanalyses are concerned with minimizing the peak heights of his arrayresponse functions within specified bands of spatial frequencies. Thereis no hint of the possibility that an important type of recordimprovement could be achieved by judicious purposeful placement of thevalleys (the zeros) of the array response functions.

The above paragraphs have cited what is believed to be the closest priorart to the present invention. The aliasing problem, in spatialfrequencies, solved by the present invention, has been mentioned in thetechnical literature as a theoretical possibility, beginning as early as1969, but no estimate has been given of the possible actual significanceof the problem. As mentioned heretofore, this would hardly have beenpossible because the effect of the problem on actual field records couldnot be measured until the problem was solved; and the previouslysuggested, obvious solution of merely decreasing geophone group spacingsenough to solve the problem was not economically and technicallyfeasible.

At this point, the seismic problem of aliasing in the spatial frequencydomain will be described verbally. More exact mathematical descriptionis given later in this specification.

The seismic recording process involves the general problem of sampling acontinuous function at regular sampling intervals, a problem whicharises in many arts. The continuous function in seismic recording is themovement of the earth, or the velocity of that movement, at points alongthe line of geophone group centers. The regular sampling interval is thegeophone group interval. It is intuitively obvious that if the wavebeing sampled is so long that the distance between any two nodes of thewave covers many of the sampling points, then the sampling will give anexcellent idea of the shape of the wave; and as a matter of mathematicalfact, it can be shown that under such conditions, the sample data cangiven an exact mathematical representation of the wave. It isintuitively evident also, that if the wave being sampled is so shortthat several oscillations of the wave occur between each two samplingpoints, then only a poor idea of the wave may be obtained. As a matterof mathematical fact, it turns out that the poorness is not utterconfusion, it is merely ambiguity. A wave that is being inadequatelysampled, by sampling points spaced too far apart, can be mathematicallyreconstructed from that sampling as being composed of one or more of aset of definite frequencies whose interrelationships depend upon thesampling interval. The ambiguity is in what proportions exist among thewaves of those definite frequencies.

The lowest frequency in that set of definite frequencies is one, whichif it were present, would be adequate ly sampled. So, another way ofstating the ambiguity is to say that wave energy which actually existedin any one member, or in several members of that set of frequencies,appears as if it were all in the lowest member, which does fall in theadequately sampled frequency range. Still another way of stating thisresult is to say that any one, or all, of the upper frequencies of theset will alias back into the range ofproperly sampled frequencies,masquerading as the one member of the set that is in the adequatelysampled range.

In the present specification, the term high spatial frequencies" refersto the high apparent spatial frequencies, (in cycles per foot) of thehorizontal components of earth waves whose spatial variation is beingattempted to be sampled by the geophone groups laid along the surface ofthe earth. Because it is not technically and economically feasible tosample the earth at close enough points to get unambiguous sampling ofall the possible shorter, higher frequency waves, when those waves arepresent they alias down into the adequately sampled frequency band.

One specific manifestation of the seismic aliasing phenomenon can bedescribed in terms of wave arrival directions. Reference may be madehere to FIG. 1, in which are shown two waves having the same truewavelength in their own directions of travel, but quite differentapparent horizontal wavelengths. The wave coming from a direction closeto the vertical has an apparent horizontal wavelength much longer thanits actual wavelength. In the example illustrated in FIG. 1,

that apparent horizontal wavelength is greater than twice the distancebetween geophone group centers, and according to the mathematics ofsampling, developed in detail later in the present specification, wavesas long as twice the distance between geophone group centers, andlonger, are quite adequately sampled. On the other hand, thehorizontally traveling wave, having the same true wavelength, but whoseapparent horizontal wavelength" is actually that true wavelength, is notadequately sampled by the represented geophone groups. That wave of tooshort wavelength, or too high spatial frequency, tends to alias down toa lower apparent spatial frequency (dependent on its actual spatialfrequency and the geophone group interval). The wave then manifestsitself in the obtained seismic record as a false wave of lower spatialfrequency, having a false apparent direction of travel closer to thevertical. The recorded evidence of the false apparent wave may then beconfused with, or interfere with, the recorded evidence of a true wavecoming from that same direction closer to the vertical. The interferencemay even be so strong as to obscure the true wave beyond recognition.

In verbal terms, this is the seismic aliasing problem in the spatialfrequency domain. The present inventors have discovered a solution tothe problem that does not require diminution of the geophone groupinterval below technically and economically reasonable limits. They havefound that the geophone spacings within the geophone groups may berelated to the geophone group intervals in just such a way that all theparticular narrow bands of spatial frequencies that would otherwisealias down into the desired, adequately sampled band of spatialfrequencies are particularly eliminated. They are eliminated by thevalleys (the zeros) of the geophone (and source)group-response-versus-spatial frequency functions.

The manner in which the aliasing spatial frequencies are eliminated, andin which other accompanying objectives are achieved, will become evidentfrom the following specification, including the drawings, whose generalcontents are listed immediately below:

BRIEF DESCRIPTIONS OF THE DRAWINGS In the drawings,

FIG. 1 is a schematic representation of a seismic exploration systemlaid out in accordance with the present invention. It shows geophone andsource spacings, and indicates how some of the geophone groupconfigurations of the present invention may be derived. It alsoillustrates some seismic waves, between which the present inventionenables discrimination.

FIG. 2A is a plot of a Fourier transform of a seismic disturbance,component amplitudes being plotted as a function of spatial frequencyllwavelength). The convenient unit of spatial frequency here is thereciprocal of the geophone group interval.

FIG. 2B is a diagram similar to FIG. 2A, showing the effect of discretesampling on the Fourier transform of the seismic signals.

FIG. 3A represents a prior an grouping of geophones called a centroidspaced seismic transducer array.

FIG. 3B is a decibel response curve of the configuration of geophones ofFIG. 3A.

FIG. 4A is an array of seismic geophones in accordance with the presentinvention.

FIG. 4B is a decibel response curve of the array of FIG. 4A. The zerosof the response curve of such an array fall at equal intervals of l/Agin the spatial frequency domain.

FIG. 5A illustrates one form of mixing of adjacent geophone groups,which have a center-to-center spacing of Ag.

FIG. SE is a representation of seismic source groups, derived byconvolution, in which the center-to-center spacing, or source groupinterval, is As.

FIG. 6 is a decibel response curve of the array shown in FIG. 5A inwhich a binomial mix has been used of the adjacent geophone groups, forrecording.

FIG. 7 is a geophone-source plot, useful in explaining the gathering ofindividual traces to make either common source gathers or commongeophone gathers. Each dot represents one time trace, time, t, isrepresented as being in a direction perpendicular to the plane of FIG.7.

FIG. 8 .is a portion of a field record recorded with conventionalgeophone groupings typical of the prior art.

FIG. 9 is a portion of a field record of the same seismic linerepresented in FIG. 8, with data collected and processed in accordancewith the present invention.

DETAILED DESCRIPTION OF PRESENT INVENTION Referring now to the drawings,FIG. 1 illustrates schematically a seismic exploration system laid outin accordance with the present invention. It shows a plurality ofgeophone groups spaced from each other and from a seismic source andconnection of each geophone group to a field recording system. Thearrangement will be described in greater detail further on in thisspecification. However, at this point, a full discussion of thetheoretical background and practical design of such a system to solvethe aliasing problem referred to above, will help those skilled in theart of seismic exploration to understand better the field and recordanalysis procedures specifically contemplated by the arrangement of FIG.1.

Let g be the distance coordinate along the line of geophone placement.If a function w of the independent variable g has a Fourier transformW(f,,), the function W 02), which is the Fourier transform of the function reconstructed from the sample values of w(g) taken at regularintervals of Ag, is given by mm) 2 (fr'l Magi, (I)

(See, e.g., Peterson, D.P. and D. Middleton: Information and Control5:279( 1962)). This relationship is illustrated in FIG. 28. It isapparent from Equation (1) and FIG. 2B that the amplitude of W,(f,,) forany value of f, satisfying lf,| VAg will be determined by the values ofW(f,) atfl, n/Ag, n 0,21, :2, The amplitudes WU, n/Ag), n =11, i2, aresaid to be aliased back to the spatial frequency j}.

Within the context of exploration seismology, FIG. 2A might representthe Fourier transform of a seismic disturbance arriving at the seismicsurvey line at some fixed record time. FIG. 28 then illustrates theeffect of sampling at an interval Ag on the Fourier transform of thereconstructed seismic disturbance. If we regard the seismic energy inthe vicinity off, as signal and the remainder as noise (in practice, weare usually most interested in the region f %Ag) it is then obvious thatthe aliasing effects due to energy in the vicinities of spatialfrequencies f, =n/Ag will tend to decrease the signal-to-noise ratio forinterpolated values of the seismic disturbance.

As previously discussed, the prior art has devised geophone and sourcearrays (or groups) which attempt to attenuate the energy at spatialfrequencies corresponding to the coherent noise waves. The geophonegroup-response-versus-spatial-frequency function of a symmetric geophonegroup may be written where G, is the sensitivity, or gain, of the n"geophone and g is the distance of the n" geophone from the array center(implying g,, O). The symmetry results from the requirements G G and g gThe number of geophones in the array is 2N if G, 0 and 2N 1 otherwise.For brevity we shall frequently refer to R(f,) as thegeophone-group-response function. A function commonly used inexploration seismology to characterize the spatial frequency attenuationproperties of a geophone array is the decibel response of the array,defined by tf 20 am l tfnll- 3) An example of a simple geophone groupfrom the prior art and its decibel response is shown in FIGS. 3A and 3B,respectively. The group consists of six equally sensitive geophoneslocated with respect to the array center as indicated in FIG. 3A. Thearray is called a centroid spaced seismic transducer array as describedin U. S. Pat. No. 3,400,783 issued to E. K. Lee, et al., on Sept. 10,1968. Assuming a geophone group interval, Ag, of 60 ft, the integralmultiples of .1/Ag will occur at the points indicated on the decibelresponse curve of FIG. 38. It will be noted that the maximum attenuationat any integral multiple of 1/Ag is approximately 23 db. It is notuncommon when using surface sources to have coherent noise waves lying50 db or more above the reflected signal. For example, the 18.4 hzcomponent of an air wave having a velocity of 1,100 ft/sec will have aspatial frequency of l/Ag 0.0167 cycles per foot, as calculated by usingthe equation: f,(spatial frequency) f,(temporal frequency)/V(velocity).4. If the amplitude of this 18.4 hz component is considerably largerthan the signal amplitude at the same record time, the aliasing effectof the coherent noise wave can easily obscure the useful information.

In FIGS. 4A and 4B are shown an array in accordance with the presentinvention, and the decibel response of that array, which has betterspatial frequency anti-alias features than those in FIG. 3B. The arrayconsists of six equally sensitive geophones located with respect to thegroup center as shown in FIG. 4A. It is assumed that the group interval,Ag, is 60 ft. If a different group interval is desired, the elementspacings in FIG. 4A must be multiplied by (Ag) desired/60. It isapparent from FIG. 43 that the transfer function of the array of FIG. 4Ahas its zeros equally spaced at intervals of l/Ag between the zeroandfirst-order major lobes. Those spatial frequencies that would give riseto aliasing effects in the neighborhood of zero spatial frequency arecompletely attenuated. This would not be possible with the array of FIG.3A since the zeros of its transfer function are not equally spaced.

The array of FIG. 4A is a relatively simple example out of the class ofall arrays having transfer functions with equally spaced zeros betweenthe major lobes. With such arrays it is always possible to adjust theelement spacing so the zeros of the transfer function will lie atspatial frequencies n/Ag, n i1, i2, in The six-element array of FIG. 4Abelongs to a sub-class of such arrays which we shall refer to as primeconvolutionary arrays. In particular, the array of FIG. 4A is theconvolution of two simple arrays having two and three elements,respectively. The two-element array has geophones located at Ag/4, Ag/4and the three-element array has geophones located at Ag/3, 0, Ag/3}, thegeophones in both arrays being equally sensitive. The array of FIG. 4Athen has geophones at locations given by the convolution of the twosimpler arrays. The geophone group-response function of the six elementarray will be equal to the product of the geophone group-responsefunctions of the twoand three-element arrays. Thus, atf, i l/Ag, thegeophones group-response functions of the twoand threeelement arrays areboth zero, so the, geophone-group-response function of the array in FIG.4A will have a second-order zero atf,= i l/Ag. This is manifested in thenormalized decibel response of FIG. 48 as an increased attenuation inthe vicinity of f, l/Ag 0.0167 cycles/ft.

Prime convolutionary arrays with better attenuation characteristics thanthe above-mentioned six-element array may be constructed as follows:Denote the array having m equally sensitive elements spaced at intervalsof Ag/m by a,. Similarly, denote the array having m equally sensitiveelements spaced at intervals of g/m by a The geophone group-responsefunction of the array 11 will be given by mot); I sin f A which hasfirst-order zeros atf,,= 1/Ag, 2/Ag, ,m llAg in the interval of 0 f, 5 mg. The array formed by convolving a with a (denoted by a,*a will have ageophone group-response function given by Sill If m m are relativelyprime, R, f,) will have firstorder zeros at f, k m lAg and k m /Ag,where k l, 2, ,(m l),and k l,2,...,(m l),in the interval f 5 m m lAg.All other zeros in this interval will be of second-order. The array inthis example will be referred to as m,*m prime convolutionary. It isconsidered to be important that the early zeros of the geophonegroup-response function be of second-order or higher in order that thenormalized decibel response have good attenuation characteristics in thevicinity of the early integral multiples of l/Ag.

There exists other types of geophone arrays whose geophonegroup-response functions have equally spaced zeros, but the primeconvolutionary arrays are particularly simple to work with as a resultof the fact that the geophone group-response function of aconvolutionary array is equal to the product of the geophonegroup-response functions of the component arrays. This fact is directlyattributable to the convolution theorem for Fourier transforms.

As discussed above, and illustrated in FIG. 4B, proper placement of thegeophones in an array leads to a high attenuation of those spatialfrequencies which give rise to aliasing effects in the vicinity of zerospatial frequency. In addition, there will be aliasing effects in thedesired passband due to spatial frequency components lying halfwaybetween the integral multiples of l/Ag. In FIG. 4B, it is seen that thedecibel response of the represented simple six-element array has maximalying approximately halfway between the integral multiples of l/Ag.

It is part of the method of the present invention to further attenuatethose spatial frequencies giving rise to aliasing effects in the highportions of the spatial frequency pass-band by combining the outputs ofnear neighboring geophone groups in an inter-trace mix. An example of ausable mixing arrangement is illustrated in FIG. 5A. The mixing is suchthat the trace value recorded by the seismic recorder, 7:,(t), at recordtime t is given by l( l-l l l+l( )y 8 where T,(t) is the output of the jgeophone group at record time (herej= i-l, i, or i+l The effect of thisparticular mixer upon the spatial frequency spectrum of the seismogramis to multiply the spectrum by the function All (fa) 0052 li t- 1] whichhas second-order zeros atf, (2k l)/2Ag, k 0, i1, i2, One may regard themixer-geophone array combination as being equivalent to a largergeophone array which is a convolution of the actual geophone array withan array of three elements spaced at an interval of Ag and havingsensitivities in the ratio 1:2: 1. The normalized decibel response ofthe convolution of the six-element array shown in FIG. 4A and the 1:2:1mixer is shown in FIG. 6.

The 1:2:1 mixer belongs to a general class of mixers whose transferfunctions have the property of being equal to zero at all of the oddhalf-integral multiples of l/Ag. A particularly useful subset of thisgeneral class is the set of mixers such that the recorded trace is givenb O y is a binomial coefficient. The transfer function of this mixer isJI -(f =[cos (wf Agfl (12) which has zeros of order 2N at allhalf-integral multiples of l/Ag. We refer to such a mixer as abinomialmixer.

The mixer contemplated in this specification is a multi-channel passiveresistive network into which the outputs of the individual geophonegroups are fed, as illustrated in FIG. 5A. The construction of suchmixers is already well-known to those skilled in the art of seismicinstrumentation. It is known that caution should be exercised inmatching the impedance to the impedance of the seismic amplifier, andsteps must be taken to minimize cross-talk among the input channels.Another method of achieving the desired inter-trace mix is to record theseismic traces, unmixed, and then perform the mixing at some later timeduring computer processing. We refer to this process ascomputer-mixing." Field-mixing is the term applied to the process ofinter-trace mixing prior to recording of the seismic traces.

In accordance with this invention, it is now possible to design geophonearrays whose geophone-groupresponse functions are zero at all earlyhalf-integral multiples of the reciprocal of the group interval, thuscombining the spatial-frequency anti-aliasing features of the geophonearrays and inter-trace mixers discussed up to this point. Illustrativeof this general class of geophone arrays is the useful set of arraysconstructed as follows: Denote the array having n equally sensitiveelements spaced at intervals of 2Ag/n by C,. Similarly, denote the arrayhaving n equally sensitive elements spaced at intervals of 2Ag/n by CThe geophonegroup-response function of array C is given by 1 (Zita 3i111 f Ag) which has first-order zeros at f,= VzAg, 1/Ag, (n l)/ 2A g inthe interval ()flsm/Q Ag. The array formed by convolving C with C willhave a geophonegroup-response function given by Rio.

my.) I n 1.) 1.) 5mg (2%) If n n are relatively prime integers, R (f,,)will have first-order zeros at f,= k n,/2Ag and k n,j2Ag, where k, 1,2,,(n 1),andk =l,2,... ,(n,=1),inthe interval j}, n nJZAg. All other zerosin this interval will be of second order. It will be noted that thegeophone array in this example is a prime convolutionary array whosegeophone-group-response function is zero at all of the 5133/half-integral multiples of the reciprocal of the group interval. In viewof this fact, it is apparent that this geophone array can be used inplace of a geophone array having a geophone-groupresponse functionhaving zeros at the early integral multiples of the reciprocal of thegroup interval and a mixer whose purpose is to cause thegeophone-groupresponse function to become zero at the earlyoddhalf-integral multiples of the reciprocal of the group intervaL Wetherefore, refer to this new type of array as a self-sufficient primeconvolution ary array.

Up to this point the present invention has been described in terms ofgeophone arrays and an intertrace mixer which operates only on thetraces of a single seismogram, or, what is also known in the art as acommon-source-group gather. In addition to the use of geophone groups toattenuate coherent noise waves the prior art has given some recognitionto the usefulness of source groups for this purpose. In view of the factthat a line of seismic data may be regarded as a function of the threeindependent variables g, s, and t, (3 being the distance coordinatealong the line of geophone placement, s the corresponding coordinate forthe sources, and t, record time), we may regard the three-dimensionalFourier transform of the seismic line as a function of the threeindependent frequency variables f,,, j}, and f,. Thus far we havediscussed the attenuation of coherent noise energy corresponding tovalues of f such that l f, l lAg. It is apparent that we may also have ameans of attenuating those portions of coherent noise wavescorresponding to spatial frequen cies such that f,1--z rAs, where As isthe previously defined source group interval. It is a part of the methodof the present invention to design and use source groups that areanalogous to the geophone groups previously described. In particular, itis desirable to use a source group whose source-group-response-versus1713 Zrr 271" sin lf Ag) s1n 2 IRAQ) spatial-frequency function (orsimply, source-groupresponse function),

ll'l" l\'l m --M (Hi) has zeros at f, k/As, k =fl, i-k In Equation (16)S is the strength of the m" individual source in the group and s is itslocation with respect to the source group center. It is desirable torequire that S.,,. S,, and s s,,.. The number of sources in the array is2M ifS O, and 2M+ 1 otherwise.

FIG. 5B illustrates a simple example of a prime convolutioanry sourcegroup of six sources of equal strength. Two adjacent, interlockingsource arrays are shown in the diagram to illustrate the meaning of As,which for this example was chosen to be equal to Ag, but this is not ageneral requirement. Just as it is considered important for R(f,) tohave second-or higherorder zeros at the early integral multiples ofl/Ag, it is also considered important that P(f,) have secondorhigher-order zeros at the early integral multiples of 1/As.

In complete analogy to the inter-trace mix among the traces of commonsource gather, described above, the method of the present inventioncontemplates a mix among the traces of a common geophone-group gather"in order further to mama; high spatial frequency components of theseismic data that would cause aliasing effects in the integral f, %As.

FIG. 7 is a diagram that is useful in describing the layout of seismicspreads and in discussing certain aspects of seismic data processing.Each point on the diagram has an s-coordinate corresponding to thelocation of the center of the source group giving rise to a seismicline. The location of the origin of the seismic line on the diagram isat g s O. The example shown in FIG. 7 is a seismic line consisting of 1412-trace seismograms, or common source-group gathers, which wererecorded with an end-on layout geometry, with As Ag. The offset of thesource group center from the near geophone array center for anyseismogram on the diagram is 2Ag.

Returning now to the description of the inter-trace mix among the tracesof a common-geophone-group gather, it should be apparent from FIG. 7what is intended. All those traces having the same g-coordinate are tobe processed by an intertrace mixer whose spa tial frequency transferfunction has zeros at all odd halt int'egral multiples of l/As. In thisway high spatial frequency components in the interval f, 9As will besubjected to an attenuation beyond that provided by the use of sourcearrays. The mathematical characteristics of this mixer are identical tothose of the previ- As in the case of geophone array design, it ispossible to design source arrays having source-group-response functionsthat are zero at all the early half-integral multiples of the reciprocalof the source group interval, thus eliminating the need for mixing thetraces of the common-geophone-group gathers at a later time. An exampleof such an array is the self-sufficient prime convolutionary geophonearray, previously described, with the equally-sensitive geophonesreplaced with sources of equal strength.

DESIGN OF GEOPHONE GROUPS ACCORDING TO THE PRESENT INVENTION From theabove considerations it will be apparent that improved geophone arraysfor field use can be designed in the following manner:

As in customary practice, the first quantity to be specified is thedistance between successive group centers, Ag. Experienced geophysicistspick this distance on the bases of both practical necessity (availablemoney, field equipment, and processing facilities) and indicatedgeologic necessity. The more pronounced are the suspected clips of thegeologic formations, the smaller must be Ag. Distances of 50 to 400 feetare representative of present practice. After Ag is specified, .thedistances between the geophones within each array need to be specified;and the simplest way (but not the best way) to do this according to thepresent invention is to space the geophones uniformly with an intervalAe Ag/m where m is the number of geophones in the group. This will causethe spatial frequency transfer function (the groupresponse-versus-spatial-frequency function) to have zeros at all theearly non-zero integral multiples of the reciprocal of the distance, Ag,between group centers, in accordance with the teachings of the previoussection. However the zeros for the simple, uniformly spaced geophonegroup will be only first order zeros, and further improvement can beobtained with second and higher order zeros. So, in accordance with theteachings of the previous section, better arrays can be designed byconvolving pairs, triplets, quadruplets, or higher ordered sets, ofsimple uniformly spaced arrays. The order of at least some of the zerosof the transfer function of the resulting geophone group will then beequal to the number of simple uniformly spaced arrays entering into theconvolution.

In accordance with the previously given theory, if two simple uniformlyspaced arrays are to be convolved, one having m elements (geophones) andthe other having n elements, the original element-to-element spacings ofthose simple arrays should be Ag/m and Ag/n respectively.

In the convolved array of FIG. 4A, which has already been described inthe preceding section, the quantities Ag/m and Ag/n must be 60/2 30 and60/3 respectively. The convolved array must then have geophones at thepositions:

:(%)(60/2)i(60/3) 15 20 35 ft l5 20 35 ft and these are the positionsshown in FIG. 4A. The arithmetic procedures for designing otherconvolved arrays are quite analogous.

AN EMBODIMENT OF THE PRESENT INVENTION IN A FIELD SYSTEM Reference isnow made again to FIG. 1 in which a seismic wave detecting and recordingsystem is shown in schematic form, with the seismic source and detectingelements spaced in accordance with the present invention, so thatseismic wave energy which would ordinarily alias back into the desiredlower spatial frequencies, can be effectively excluded from the recordedwaves.

Geophone groups of the sort already described in connection with FIG. 4Aare shown in FIG. 1, interdigited with their neighboring geophonegroups. The groups in FIG. 1 are shaded alternately black and white toshow clearly which geophones belong to each group. This brings out oneof the most interesting features of prime convolutionary groups. Thegeophones of each group overlap the geophones of its neighboring groupin just such a manner that a uniformly-spaced set of locations iscompletely filled, each with one geophone (and only one). The relativelysimple, 2*3 prime convolutionary groups shown in FIG. 1 have overlapsconsisting of only one geophone apiece from each of two neighboringgroups; higher-ordered prime convolutionary groups have complicatedoverlap sequences.

In the upper left hand portion of FIG. 1, the diagram is intended toindicate how the geophone positions could have been derived byconvolutions of 2-element and 3-element component arrays. Dots 10a and11a represent component arrays of m elements whose element-to-elementspacing is Ag/m. Dots 10b and 11b represent component arrays of nelements whose element-to-element spacing is Ag/n. In the convolutionoperation, for each element position 10a or 11a, there is substituted agroup of element positions 10b or 11b respectively. With the particulararrays shown for each original component array of two positions, thereresults an array having six positions. The respective positions,obtained by convolution are indicated by the dashed liner leading to thefinal geophone positions, 10 and l l.

The common electrical leads, such as 100 and 11c are shown connected atcenter positions of the geophone groups.

Geophones l0 and 11 along the line of survey are actuated by seismicwaves generated by a seismic source 14, which in the representedschematic embodiment is shown as an eccentric weight 12 mounted forrotation on a base 13 by a prime mover, such as electric motor 16.Control of the rate of vibration is through the speed of motor 16, bycontroller 18.

As indicated in phantom, the spacing As between succeeding sources alonghorizontal coordinate s is the same as the geophone group centerspacing, Ag, along the horizontal coordinate g.

The leads from the geophone groups in FIG. I go to mixing circuits 20,21, and 22, whose components are symbolized here as operationalamplifiers, although as mentioned hereinbefore, the mixers may bepassive resistive networks. The outputs of the mixers go through leads25, directly, or indirectly through other amplifiers not shown, torecording heads 27 to generate traces on record 30 driven in timesynchronism by motor 32. In actual present practice, the leads 25 may gointo a unit which records digital signals directly on magnetic tape. Asexplained in the foregoing parts of this specification, the overallpurpose of an entire system, such as that represented in FIG. 1 is toavoid the recording of certain sets of spatial frequencies, which arethe halfinteger multiples of the reciprocal of the geophone groupinterval, and the half-integer multiples of the reciprocal of the source(group) interval spacing.

ACTUAL DATA SHOWING IMPROVEMENT ACHIEVABLE BY PRESENT INVENTION In orderto show the kind of improvement in seismic records that can be achievedusing the method of the present invention, FIGS. 8 and 9 are presented.

FIG. 8 is a stacked time section produced by conventional recording andcomputer-processing methods. The sources used were of the nonexplosivesurface type and the equivalent source array was a linear set of 21sources spaced at an interval of 13 ft with relative strengths in theratio l,2,4,6,7,8,8,8,8,8,8,8,8,8 ,8,8,7,6,4,2,1 The equivalent geophonearray was a set of detectors spaced at an interval of l3 ft withrelative sensitivities in the ratio1,1,1,l,2,2,2,2,3,3,3,3,2,2,2,2,1,l,l,l In field practice, such ageophone array is realized by placing one, two, or three equallysensitive geophones at the detector locations requiring multiplesensitivity. The source group interval was equal to twice the geophonegroup interval (As 312 ft, Ag= 156 ft).

FIG. 9 is a stacked time section produced by the method of the presentinvention. The seismic survey line was the same as that for FIG. 8. Thesource used was again of the nonexplosive surface type and theequivalent source array was a 3*11 prime convolutionary array. Thegeophone array was a 4*9 prime convolutionary array. A 1:4:6:4:lfield-binomial-mix was applied to the traces of thecommon-source-groupgathers and a 1:221 computer-binomial-mix was appliedto the traces of the common-geophone-group-gathers. The source groupinterval was equal to the geophone group interval (As Ag =165 ft).

The remarkable superiority of FIG. 9 to FIG. 8, in depicting subsurfacestratal configurations, is immediately obvious upon inspection; and thecomparison illustrates representatively the superiority of the method ofthe present invention to conventional methods.

SUMMARY In summary, the method of the present invention effectivelysolves the problem of alias energy due to improper or inadequatesampling of the seismic waves resulting in higher spatial frequenciesfrom surface waves or noise being recorded as lower spatial frequenciesfrom seismic reflections. As particularly distinguished from previouslyproposed solutions to this problem, either using an uneconomic orimpractical number of sources or geophones, or attempting to suppressall frequencies above a given passband of useful seismic frequencies, itwill now be understood, in accordance with the teachings of thisapplication, that correct placement of seismic elements (sources, and/orgeophones) within their respective groups, causes the group response tobe controlled so that just the right set of narrow frequency bands inthe spatial frequency spectrum is effectively excluded from the recordedand processed seismic waves.

While various modifications in apparatus and procedure will becomeapparent from the foregoing description, all such modifications comingwithin the scope of the claims are intended to be included.

I claim:

1. In a method of seismic geophysical prospecting, which includesgenerating seismic waves by seismic energy sources horizontally spacedalong a line of survey, and detecting subsequent reflections of thegenerated seismic waves from subsurface strata, by a plurality of groupsof geophones horizontally spaced along said line of survey adjacent thesurface of the earth, each group of geophones producing a seismic traceto be displayed with other traces, side-by-side, to form a seismicrecord, in which method the prior art attempted to suppress broad bandsof undesired, higher spatial frequencies, to minimize the interferingeffects of spatial frequencies above the desired pass band, theimprovement of more effectively suppressing only those particularspatial frequencies that would alias" back into said desired pass band,which comprises positioning at least one of the two kinds of seismicelements, consisting of geophones and seismic sources, so that theirrespective group-response-versus-spatialfrequency functions suppress thealiasing frequencies, by performing the following steps:

a. placing a plurality of individual geophones within each of saidgeophone groups at horizontal positions, denoted in terms of thehorizontal coordinate, g, along said line of survey, said positionsbeing the convolved positions from within two uniformly spaced componentarrays, the first array having m geophone positions and the second arrayhaving n geophone positions, where m and n are integers, and theindividual, uniform, geophoneto-geophone spacings in said firstcomponent array and said second component array, are respectively, Ag/mand Ag/n, where Ag is the distance between successive geophone groupcenters along said line of survey,

. placing individual sources at a plurality of horizontal positionsdenoted in terms of the horizontal coordinate s, along said line ofsurvey, the distance between individual sources, As, being equal to Ag,c. energizing at least one of said sources, and

. adding the traces from at least one set of three neighboring geophonegroups that have received waves caused by a common source, usingbinomial weighting, the central trace being given a weight twice that ofeach of the two adjacent traces so that the effective geophonegroup-response-versus-spatial-frequency function is substantially zeroat the half-integral multiples of llAg.

2. In a method of seismic geophysical prospecting, which includesgenerating seismic waves by seismic energy sources horizontally spacedalong a line of survey, and detecting subsequent reflections of thegenerated seismic waves from subsurface strata, by a plurality of groupsof geophones horizontally spaced those particular spatial frequenciesthat would alias back into said desired pass band, which comprisespositioning at least one of the two kinds of seismic elements,consisting of geophones andiseismic sources, so that their respectivegroup-response-versus-spatial-t frequency functions suppress thealiasing frequencies,

' by performing the following steps:

a. placing a plurality of individual sources within each of a set ofsource groups at horizontal positions, denoted in terms of thehorizontal'coor dinate, s, along said line of survey, said positionswithin each group being the convolved positions from within twouniformly spaced component arrays, the first array having 1; sourcepositions and the second array having q source positions, where p and qare integers, and the individual, uniform, source-to-source spacings insaid first component array and said second component array, arerespectively, As/p and As/q, where As is the distance between successivesource grouptcenters along said line of survey, r

. placing a plurality of individual geophones within each of saidgeophone groups at horizontal posi-' tions, denoted in terms of thehorizontal coordinate, g, along said line of survey, said positionsbeing the convolved positions from within two uniformly spaced componentarrays, the first array having m geophone positions and the second arrayhaving n geophone positions, where m and n are integers, and theindividual, uniform, geophoneto-geophone spacings in said firstcomponent array and said second component array, are respectively, Ag/mand Ag/n, where Ag is the distance between successive geophone groupcenters along said line of survey, and Ag As, c. energizing separatelyand sequentially at least three neighboring ones of said source groups,

adding the traces from at least one set of three neighboring geophonegroups that have received waves caused by a common source group, usingbinomial weighting, the central trace being given a weight twice that ofeach of the two adjacent traces, and e. adding at least one set of threetraces that have been received by one geophone group from the actuationof three neighboring source groups, using binomial weighting, thecentral trace being given a weight twice that of each of the twoadjacent traces so that both the effectivesource-groupresponse-versus-spatial-frequency function and the effectivegeophone-group-response-versusspatial-frequency function aresubstantially zero at the half-integral multiples of l/Ar l/Ag. 3. In amethod of seismic geophysical prospecting, which includes generatingseismic waves by seismic energy sources horizontally spaced along a lineof survey, and detecting subsequent reflections of the generated seismicwaves from subsurface strata, by a plurality of groups of geophoneshorizontally spaced along said line of survey adjacent the surface ofthe t earth, each group of geophones producing a seismic trace to bedisplayed with other traces, side-'by-side, to form a seismic record, inwhich method the prior art attempted to suppress broad bands ofundesired, higher spatial frequencies, to minimize the interferingeffects of spatial frequencies above the desired pass band, the

' improvement of more effectively suppressing only those particularspatial frequencies that would alias back into said desired pass band,which comprises positioning at least'one of two kinds of seismicelements, consisting of geophones and seismic sources, so that theirrespective group-responseaversus-spatial-frequency functions suppressthe aliasing frequencies, by performing the following steps:

a. positioning pluralities of one of said kinds of seismic elements ingroups along said line of sur- 'vey so that the horizontal intervalbetween the centers of'adjacent groups is Ae,

. positioning the individual elements of said one kind of seismicelement within its respective group so that the respectivegroupresponse-versus-spatial-frequency function becomes substantiallyzero at the early, integral multiples ofl/Ae, t

c. mixing seismic traces from near neighboring' groups of said one kindof seismic element and a common single group of said other kinds ofseismic elements, so that the effective said respectivegroup-response-versus-spatial-frequency function becomes substantiallyzero also at the odd, half-integral multiples of l/Ae.

' 4. The method of claim 3 in which said' one of said kinds of seismicelements is a geophone and said other of said kinds of seismic elementsis a source.

5. The method of claim 3 in which said one of said kinds of seismicelements is a source and the other of said kinds of seismic elements isa geophone.

6. In a method of seismic geophysical prospecting, which includesgenerating seismic waves by seismic energy sources horizontally spacedalong a line of survey, and detecting subsequent reflections of thegenerated seismic waves from subsurface strata, by a plurality of groupsof geophones horizontally spaced along said line of survey adjacent thesurface of the earth, each group of geophones producing a seismic traceto be displayed with other traces, side-by-side, to form a seismicrecord, in which method the prior art attempted to suppress broad bandsof undesired, higher spatial frequencies, to minimize the interferingeffects of spatial frequencies above the desired pass band, theimprovement of more effectively suppressing only those particularspatial frequencies that would alias back into said desired pass band,which comprises positioning at least one of two kinds of seismicelements, consisting of geophones and seismic sources, so that theirrespective group-response-versus-spatial-frequency functions suppressthe aliasing frequencies, by performing the following steps:

a. positioning pluralities of one of said kinds of seismic elements ingroups along said line of survey so that the horizontal interval betweenthe centers of adjacent groups is Ae,

b. positioning the individual elements of said one kind of seismicelement within its respective group so that the respectivegroup-response-versus-spatial-frequency function becomes substantiallyzero at all the early half-integral multiples of 1 Me.

7. The method of claim 6 in which said one of said kinds of seismicelements is a geophone.

8. The method of claim 6 in which said one of said kinds of seismicelements is a source.

9. The method of claim 4 with the additional step of positioningpluralities of sources in groups along said line of survey, theindividual sources within the source groups being positioned so that thesource groupresponse-versus-spatial-frequency function becomessubstantially zero at the early integral multiples of l/As, thereciprocal of the source group interval.

10. The method of claim 4 with the additional step of mixing seismictraces from adjacent source groups and a common geophone group in abinomial mix, so that the effective sourcegroup-response-versus-spatialfrequency function becomes substantiallyzero at the odd, half-integral multiples of 1/As, the reciprocal of thesource group interval.

11. The method of claim 7 with the additional step of positioningpluralities of sources in groups along said line of survey, theindividual sources within the source groups being positioned so that thesource groupresponse-versus-spatial-frequency function becomessubstantially zero at the early integral multiples of l/As, thereciprocal of the source group interval.

12. The method of claim 7 with the additional step of mixing seismictraces from adjacent source groups and a common geophone group in abinomial mix, so that the effective sourcegroup-response-versus-spatialfrequency function becomes substantiallyzero at the odd, half-integral multiples of HM, the reciprocal of thesource group interval.

13. The method of claim 5 with additional step of positioningpluralities of geophones in groups along said line of survey, theindividual geophones within the geophone groups being positioned so thatthe geophone group-response-versus-spatial-frequency function becomessubstantially zero at the early integral multiples of l/Ag, thereciprocal of the geophone group interval.

14. The method of claim 5 with the additional step of mixing seismictraces from adjacent geophone groups and a common source group in abinomial mix, so that the effective geophonegroup-response-versus-spatialfrequency function becomes substantiallyzero at the odd, half-integral multiples of l/Ag, the reciprocal of thegeophone group interval.

15. The method of claim 8 with the additional step of positioningpluralities of geophones in groups along said line of survey, theindividual geophones within the geophone groups being positioned so thatthe geophone group-response-versus-spatial-frequency function becomessubstantially zero at the early integral multiples of 1 /Ag, thereciprocal of the geophone group interval.

16. The method of claim 8 with the additional step of mixing seismictraces from adjacent geophone groups and a common source group in abinomial mix, so that the effective geophonegroup-response-versus-spatialfre uenc function becomes substantiallyzero at the 0d hal -1ntegral multiples of l/Ag, the reciprocal of thegeophone group interval.

1. In a method of seismic geophysical prospecting, which includes generating seismic waves by seismic energy sources horizontally spaced along a line of survey, and detecting subsequent reflections of the generated seismic waves from subsurface strata, by a plurality of groups of geophones horizontally spaced along said line of survey adjacent the surface of the earth, each group of geophones producing a seismic trace to be displayed with other traces, side-by-side, to form a seismic record, in which method the prior art attempted to suppress broad bands of undesired, higher spatial frequencies, to minimize the interfering effects of spatial frequencies above the desired pass band, the improvement of more effectively suppressing only those particular spatial frequencies that would ''''alias'''' back into said desired pass band, which comprises positioning at least one of the two kinds of seismic elements, consisting of geophones and seismic sources, so that their respective group-responSe-versus-spatial-frequency functions suppress the aliasing frequencies, by performing the following steps: a. placing a plurality of individual geophones within each of said geophone groups at horizontal positions, denoted in terms of the horizontal coordinate, g, along said line of survey, said positions being the convolved positions from within two uniformly spaced component arrays, the first array having m geophone positions and the second array having n geophone positions, where m and n are integers, and the individual, uniform, geophone-to-geophone spacings in said first component array and said second component array, are respectively, Delta g/m and Delta g/n, where Delta g is the distance between successive geophone group centers along said line of survey, b. placing individual sources at a plurality of horizontal positions denoted in terms of the horizontal coordinate s, along said line of survey, the distance between individual sources, Delta s, being equal to Delta g, c. energizing at least one of said sources, and d. adding the traces from at least one set of three neighboring geophone groups that have received waves caused by a common source, using binomial weighting, the central trace being given a weight twice that of each of the two adjacent traces so that the effective geophone group-response-versus-spatial-frequency function is substantially zero at the half-integral multiples of 1/ Delta g.
 1. In a method of seismic geophysical prospecting, which includes generating seismic waves by seismic energy sources horizontally spaced along a line of survey, and detecting subsequent reflections of the generated seismic waves from subsurface strata, by a plurality of groups of geophones horizontally spaced along said line of survey adjacent the surface of the earth, each group of geophones producing a seismic trace to be displayed with other traces, side-by-side, to form a seismic record, in which method the prior art attempted to suppress broad bands of undesired, higher spatial frequencies, to minimize the interfering effects of spatial frequencies above the desired pass band, the improvement of more effectively suppressing only those particular spatial frequencies that would ''''alias'''' back into said desired pass band, which comprises positioning at least one of the two kinds of seismic elements, consisting of geophones and seismic sources, so that their respective group-responSe-versus-spatial-frequency functions suppress the aliasing frequencies, by performing the following steps: a. placing a plurality of individual geophones within each of said geophone groups at horizontal positions, denoted in terms of the horizontal coordinate, g, along said line of survey, said positions being the convolved positions from within two uniformly spaced component arrays, the first array having m geophone positions and the second array having n geophone positions, where m and n are integers, and the individual, uniform, geophone-to-geophone spacings in said first component array and said second component array, are respectively, Delta g/m and Delta g/n, where Delta g is the distance between successive geophone group centers along said line of survey, b. placing individual sources at a plurality of horizontal positions denoted in terms of the horizontal coordinate s, along said line of survey, the distance between individual sources, Delta s, being equal to Delta g, c. energizing at least one of said sources, and d. adding the traces from at least one set of three neighboring geophone groups that have received waves caused by a common source, using binomial weighting, the central trace being given a weight twice that of each of the two adjacent traces so that the effective geophone group-response-versus-spatial-frequency function is substantially zero at the half-integral multiples of 1/ Delta g.
 2. In a method of seismic geophysical prospecting, which includes generating seismic waves by seismic energy sources horizontally spaced along a line of survey, and detecting subsequent reflections of the generated seismic waves from subsurface strata, by a plurality of groups of geophones horizontally spaced along said line of survey adjacent the surface of the earth, each group of geophones producing a seismic trace to be displayed with other traces, side-by-side, to form a seismic record, in which method the prior art attempted to suppress broad bands of undesired, higher spacial frequencies, to minimize the interfering effects of spatial frequencies above the desired pass band, the improvement of more effectively suppressing only those particular spatial frequencies that would ''''alias'''' back into said desired pass band, which comprises positioning at least one of the two kinds of seismic elements, consisting of geophones and seismic sources, so that their respective group-response-versus-spatial-frequency functions suppress the aliasing frequencies, by performing the following steps: a. placing a plurality of individual sources within each of a set of source groups at horizontal positions, denoted in terms of the horizontal coordinate, s, along said line of survey, said positions within each group being the convolved positions from within two uniformly spaced component arrays, the first array having p source positions and the second array having q source positions, where p and q are integers, and the individual, uniform, source-to-source spacings in said first component array and said second component array, are respectively, Delta s/p and Delta s/q, where Delta s is the distance between successive source group centers along said line of survey, b. placing a plurality of individual geophones within each of said geophone groups at horizontal positions, denoted in terms of the horizontal coordinate, g, along said line of survey, said positions being the convolved positions from within two uniformly spaced component arrays, the first array having m geophone positions and the second array having n geophone positions, where m and n are integers, and the individual, uniform, geophone-to-geophone spacings in said first component array and said second component array, are respectively, Delta g/m and Delta g/n, where Delta g is the distance between successive geophone group centers along said line of survey, and Delta g Delta s, c. energizing separately and sequentially at least three neighboring ones of said source groups, d. adding the traces from at least one set of three neighboring geophone groups that have received waves caused by a common source group, using binomial weighting, the central trace being given a weight twice that of each of the two adjacent traces, and e. adding at least one set of three traces that have been received by one geophone group from the actuation of three neighboring source groups, using binomial weighting, the central trace being given a weight twice that of each of the two adjacent traces so that both the effective source-group-response-versus-spatial-frequency function and the effective geophone-group-response-versus-spatial-frequency function are substantially zero at the half-integral multiples of 1/ Delta s 1/ Delta g.
 3. In a method of seismic geophysical prospecting, which includes generating seismic waves by seismic energy sources horizontally spaced along a line of survey, and detecting subsequent reflections of the generated seismic waves from subsurface strata, by a plurality of groups of geophones horizontally spaced along said line of survey adjacent the surface of the earth, each group of geophones producing a seismic trace to be displayed with other traces, side-by-side, to form a seismic record, in which method the prior art attempted to suppress broad bands of undesired, higher spatial frequencies, to minimize the interfering effects of spatial frequencies above the desired pass band, the improvement of more effectively suppressing only those particular spatial frequencies that would ''''alias'''' back into said desired pass band, which comprises positioning at least one of two kinds of seismic elements, consisting of geophones and seismic sources, so that their respective group-response-versus-spatial-frequency functions suppress the aliasing frequencies, by performing the following steps: a. positioning pluralities of one of said kinds of seismic elements in groups along said line of survey so that the horizontal interval between the centers of adjacent groups is Delta e, b. positioning the individual elements of said one kind of seismic element within its respective group so that the respective group-response-versus-spatial-frequency function becomes substantially zero at the early, integral multiples of 1/ Delta e, c. mixing seismic traces from near neighboring groups of said one kind of seismic element and a common single group of said other kinds of seismic elements, so that the effective said respective group-response-versus-spatial-frequency function becomes substantially zero also at the odd, half-integral multiples of 1/ Delta e.
 4. The method of claim 3 in which said one of said kinds of seismic elements is a geophone and said other of said kinds of seismic elements is a source.
 5. The method of claim 3 in which said one of said kinds of seismic elements is a source and the other of said kinds of seismic elements is a geophone.
 6. In a method of seismic geophysical prospecting, which includes generating seismic waves by seismic energy sources horizontally spaced along a line of survey, and detecting subsequent reflections of the generated seismic waves from subsurface strata, by a plurality of groups of geophones horizontally spaced along said line of survey adjacent the surface of the earth, each group of geophones producing a seismic trace to be displayed with other traces, side-by-side, to form a seismic record, in which method the prior art attempted to suppress broad bands of undesired, higher spatial frequencies, to minimize the interfering effects of spatial frequencies above the desired pass band, the improvement of more effectively suppressing only those particular spatial frequencies that would ''''alias'''' back into said desired pass band, which comprises positioning at least one of two kinds of seismic elements, consisting of geophones aNd seismic sources, so that their respective group-response-versus-spatial-frequency functions suppress the aliasing frequencies, by performing the following steps: a. positioning pluralities of one of said kinds of seismic elements in groups along said line of survey so that the horizontal interval between the centers of adjacent groups is Delta e, b. positioning the individual elements of said one kind of seismic element within its respective group so that the respective group-response-versus-spatial-frequency function becomes substantially zero at all the early half-integral multiples of 1/ Delta e.
 7. The method of claim 6 in which said one of said kinds of seismic elements is a geophone.
 8. The method of claim 6 in which said one of said kinds of seismic elements is a source.
 9. The method of claim 4 with the additional step of positioning pluralities of sources in groups along said line of survey, the individual sources within the source groups being positioned so that the source group-response-versus-spatial-frequency function becomes substantially zero at the early integral multiples of 1/ Delta s, the reciprocal of the source group interval.
 10. The method of claim 4 with the additional step of mixing seismic traces from adjacent source groups and a common geophone group in a binomial mix, so that the effective source group-response-versus-spatial-frequency function becomes substantially zero at the odd, half-integral multiples of 1/ Delta s, the reciprocal of the source group interval.
 11. The method of claim 7 with the additional step of positioning pluralities of sources in groups along said line of survey, the individual sources within the source groups being positioned so that the source group-response-versus-spatial-frequency function becomes substantially zero at the early integral multiples of 1/ Delta s, the reciprocal of the source group interval.
 12. The method of claim 7 with the additional step of mixing seismic traces from adjacent source groups and a common geophone group in a binomial mix, so that the effective source group-response-versus-spatial-frequency function becomes substantially zero at the odd, half-integral multiples of 1/ Delta s, the reciprocal of the source group interval.
 13. The method of claim 5 with additional step of positioning pluralities of geophones in groups along said line of survey, the individual geophones within the geophone groups being positioned so that the geophone group-response-versus-spatial-frequency function becomes substantially zero at the early integral multiples of 1/ Delta g, the reciprocal of the geophone group interval.
 14. The method of claim 5 with the additional step of mixing seismic traces from adjacent geophone groups and a common source group in a binomial mix, so that the effective geophone group-response-versus-spatial-frequency function becomes substantially zero at the odd, half-integral multiples of 1/ Delta g, the reciprocal of the geophone group interval.
 15. The method of claim 8 with the additional step of positioning pluralities of geophones in groups along said line of survey, the individual geophones within the geophone groups being positioned so that the geophone group-response-versus-spatial-frequency function becomes substantially zero at the early integral multiples of 1/ Delta g, the reciprocal of the geophone group interval. 